Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, you can assume that the convex region is described by intersection of closed half spaces? In the end, it will be great if you can generalize the problem to higher dimensions.

Paramaterize the line, and take the equation for each convex half-space and replace the variables with their paramaterized version, so you have a linear inequality in one variable. Checking whether a system of linear inequalities in one variable have a solution is easy.
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Will SawinMay 10 '13 at 6:03

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That's probably optimal by default, although I imagine that preprocessing on the convex region would allow faster query processing.
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Ricky DemerMay 10 '13 at 6:59